Parallel transport for Higgs bundles over p-adic curves
Daxin Xu
TL;DR
The paper constructs and analyzes a p-adic Simpson-type framework for curves of genus at least 2, linking finite-dimensional representations of the geometric étale fundamental group to Higgs bundles whose underlying bundles admit strongly semistable degree-zero reductions. By developing Higgs crystals, the Higgs envelope, and a twisted inverse image formalism within the Faltings topos, it defines a global functor $H_{ ext{X}, ext{Exp}}$ from $f C$-representations to Higgs bundles and a quasi-inverse $V_{ ext{X}, ext{Exp}}$ back to representations, with careful descent arguments ensuring independence from choices. It proves that the image Higgs bundles are semistable of degree zero, and that the constructions are compatible with Deninger–Werner theory, yielding an abelian, closed-under-extensions subcategory $ ext{HB}^{ ext{pDW}}_{ ext{X,Exp}}(X_{f C})$ that encapsulates the pDW phenomenon. The work provides a concrete realization of how true $p$-adic representations arise from, and recover, Higgs data in the p-adic Simpson landscape, and offers substantial evidence toward Faltings’ conjecture in the curve case through explicit descent and parallel transport.
Abstract
Faltings conjectured that under the p-adic Simpson correspondence, finite dimensional p-adic representations of the geometric étale fundamental group of a smooth proper p-adic curve X are equivalent to semi-stable Higgs bundles of degree zero over X. In this article, we establish, over a p-adic curve of genus $g\ge 2$, an equivalence between these representations and Higgs bundles, whose underlying bundles potentially admit a strongly semi-stable reduction of degree zero. We show that these Higgs bundles are semi-stable of degree zero and investigate some evidence for the aforementioned conjecture.